Take a bunch of particles (can be point particles) at ${\bf{x}}_i(t)$ and of mass $m_i>0$. The gravitational field is defined by the Poisson equation $$ \nabla^2 \Phi({\bf{x}},t) = 4 \pi G \rho({\bf{x}},t)=4 \pi G\sum_i m_i \delta({\bf{x}}-{\bf{x}}_i(t)) $$ Now, this creates a coupling between the point particles, but it is not a "dynamical" coupling: given the mass configuration $\rho({\bf{x}},t)$ the field $\Phi$ is determined everywhere instantaneously. This is because the Poisson equation does not contain any temporal derivative: $\Phi({\bf{x}},t)$ is not a dynamical object. Strictly speaking, $\Phi({\bf{x}},t)$ is not even needed and we may work directly with the force and the set of ordinary differential equations $m_i \ddot{{\bf x}}_i = {{\bf F}}_i$.
I wonder if the following modification of Newton's gravity has ever been taken into account (I guess so, as it looks quite natural): we promote $\Phi$ to the state of dynamical field by substituting the Laplacian with the wave operator: $$ \frac{\partial^2}{c^2 \, \partial t^2} \Phi({\bf{x}},t) -\nabla^2 \Phi({\bf{x}},t) = -4 \pi G \rho({\bf{x}},t)= - 4 \pi G\sum_i m_i \delta({\bf{x}}-{\bf{x}}_i(t)) $$ As $c\rightarrow \infty$ we recover Newton. Is there a natural action principle for the field $\Phi$ and the point masses degrees of freedom? In this way it should be possible to obtain the equation of motion for the masses: since we modified the Poisson equation now I am not sure that it simply reads $ \ddot{{\bf x}}_i \propto -\nabla{\Phi}$.
